The study of how hydrodynamic behavior emerges from the unitary evolution of many-particle systems is a central problem in non-equilibrium statistical mechanics. Quantum simulation, proposed by Feynman, offers a powerful tool to investigate this, with both analogue and digital approaches. Digital quantum simulation, while promising for its flexibility in simulating different Hamiltonians, faces challenges due to noise and limited qubit coherence times in near-term devices. One-dimensional interacting quantum spin systems, such as the spin-XXZ chain, serve as crucial models in many-body physics. This work focuses on the high-temperature anomalous hydrodynamics of the spin-XXZ chain at the isotropic point, a topic of significant research interest. Recent work has revealed high-temperature spin super-diffusion at this point, with a current scaling consistent with KPZ universality. While experimental and analogue simulation evidence supports this KPZ scaling, there's a need for further investigation, particularly using digital quantum simulation on noisy near-term devices. This paper aims to bridge this gap by employing error mitigation techniques to extract reliable quantitative results from a digital quantum simulation on a real quantum device.

The paper reviews existing literature on quantum simulation, highlighting both analogue and digital approaches and their respective strengths and weaknesses. It discusses previous digital simulations of one-dimensional quantum spin systems, emphasizing the limitations imposed by small system sizes and short simulation times. The authors then delve into the theoretical understanding of high-temperature spin transport in the XXZ model, citing studies that have observed super-diffusion and conjectured KPZ universality at the isotropic point. The literature review highlights previous experimental and analogue simulation results that confirm this anomalous scaling, setting the stage for the current study's investigation using a digital quantum simulation approach.

The research employs a digital quantum simulation using the ibmq-montreal 27-qubit device. The authors utilize a Trotterized version of the XXZ model, which is advantageous for mitigating Trotter errors that plague continuous-time simulations. They employ a novel initial state preparation method proposed by Richter and Pal, which utilizes specially tailored pseudo-random states generated by a shallow-depth random circuit. This circuit is designed to prepare an initial state with a single spin excitation on a homogeneous background. The discrete-time dynamics of the spin auto-correlation function is then simulated using a first-order Trotter decomposition. A zero-noise extrapolation error mitigation strategy is applied. To increase the number of data points for the power-law fit, the authors implement a 'weaving' technique, which artificially increases the time resolution. Classical simulations are used to validate the approach and demonstrate the independence of the transport exponents from the Trotter step size. The power-law scaling behavior is extracted from the quantum simulation results by fitting a power law to a specific time regime where the quantum results closely match classical results and show little error.

The main finding is the successful extraction of the KPZ anomalous exponent from the digital quantum simulation on the ibmq-montreal device. The experimental results closely track classical simulations across two decades of time evolution, confirming the emergence of hydrodynamic scaling. The extracted exponent, approximately -0.644, is within 3.40% of the expected value of -2/3 for the clean model, demonstrating the validity of the KPZ scaling conjecture. The authors also demonstrate the restoration of diffusion (exponent approximately -0.505, within 1.00% of the expected -1/2) upon applying a staggered external field, which breaks the integrability of the system. The analysis accounts for device noise and errors (readout error of 2.26% for qubit 0 and average CNOT error of 1.12%). The successful implementation of the weaving technique significantly enhances the amount of available data points for robust fitting and analysis, resulting in reliable extraction of the transport exponent.

The results provide strong evidence that KPZ scaling and the restoration of diffusion through integrability breaking can be successfully simulated on near-term quantum devices. The use of pseudo-random initial states appears crucial for extracting infinite-temperature transport exponents in noisy environments. The resilience of these states to unital channels, such as dephasing, likely contributes to the success of the simulation. The discrete-time model, free from Trotter errors, enables simulations across long timescales. This work is, to the authors' knowledge, the first successful extraction of transport exponents of an interacting quantum system on a digital quantum device. The findings are consistent with experimental results from various platforms, further validating the accuracy of the digital simulation approach. As quantum hardware improves, this method promises to unlock access to high-temperature transport studies in various models.

This study successfully demonstrated the digital quantum simulation of Kardar-Parisi-Zhang (KPZ) scaling and the restoration of diffusion via integrability breaking in the spin-XXZ model on a noisy intermediate-scale quantum (NISQ) device. The use of pseudo-random initial states and a discrete-time Trotterization proved highly effective. Future work could explore the precise interplay between noise and pseudo-typical initial states, investigating the robustness of this approach to different noise models. The methodology presented here can be extended to other interacting quantum many-body models, enabling the study of transport phenomena in regimes inaccessible to classical numerical methods.

The study is limited by the capabilities of the current NISQ device. The presence of device noise and errors, while mitigated, still affects the accuracy of the results. The simulation is limited to a specific range of time evolution, dictated by the device's coherence time and the error mitigation strategy. The initial state preparation, although effective, might not perfectly represent a fully Haar random state, potentially influencing the results. Further research is needed to explore the effect of these limitations more thoroughly.